What is the reason for pedal straps? They look as if they are more awkward and I can't see what the benefits are.
I can't remember the last time my feet slipped off my mountain bike, so why would they on a road bike?
There are a number of reasons for using either toe-clips (pedal straps) or clipless pedals:
The answer "so you can pull up on the pedals" makes good sense, but I don't buy it. I don't pull up on the pedals unless I'm climbing a steep, sudden hill. I suspect many, many cyclists don't pedal with full efficiency on the flats. (This is my gut feeling, I have no proof or numbers on this. I remember seeing a study that agreed with this, but I also recall that it was limited in such a way as to cast doubt on the data.)
The reason I use clips and straps is that a lot of energy is expended simply keeping the balls of your feet centered on the pedals. If you don't believe me, try pedaling exclusively with clips and straps until you're used to them, then remove them. You'll find that your feet keep slipping off the pedals. This demonstrates that you were expending effort to keep your feet on-target.
(The only reason I don't use clipless pedals and shoes -- which are a much more elegant way to achieve the same effect clips and straps are going for -- is that I don't want to have to deal with an extra pair of shoes, and I want comfortable shoes when I get off the bike.)
An experienced cyclist will make constant, small adjustments to keep their feet in the optimal location on the pedals. Clips and straps keep the cyclist's feet in the best places to deliver the most power to the pedals (the balls of the feet), saving the cyclist that effort. (I have a regrettable habit of pedaling on the arches of my feet, which clips and straps correct.) As a bonus, the straps are there when you turn a corner and then see a sudden hill you didn't have a chance to build up momentum to climb.
Edit: I'm at least half retro-grouch touring cyclist, so take this with a grain of salt! I'm a non-racer, saddle-level-with-the-handlebars-for-comfort cyclist.
The reason for using pedal straps is to allow you to transfer power from your legs to the pedals while both pushing down and pulling up. If your pedals do not have straps you can only transfer power to the pedals when pushing down on them.
While road cycling you typically don't need to take your feet off of the pedals as often as you do when mountain biking. By having pedal straps or "clip-less" pedals you can transfer power from your legs to the pedals for the full pedal stroke.
I practice XC, DH, Touring and Urban Commuting types of cycling, and have been using cleated shoes since 2002. By now I recognize several advantages of foot retention systems, even though I haven't used pedal straps. I will share some of the advantages I have found, as I assume they are the same as correctly used straps.
For XC on rough terrain and DH, they simply offer a lot of control, prevent injury by avoiding foot slip-off. They also allow for power-surge pedalling when acceleration after a slowing obstacle.
For Touring It helps me to keep the foot on the right spot when I'm tired. It also helps me a lot to have the possibility of pulling the pedals up. Of course I'm tired, but pulling uses a different set of muscles, so at least one minute of pulling-only can relieve some strain on the more demanded muscles. The idea is to alternate one minute of pulling only with 10-15 minutes of circular pedalling.
There is less benefit for urban commuting, but for parts of the city where I need to stop repeatedly, having the cleats lets me place the pedal right at the beginning of the down stroke so I'm ready for the next sprint in just one no-need-to-think movement.
At the time of writing this answer, I have recently moved to a different country and could't bring my bike. I bought one locally but it still lacks the necessary customization, and I miss my cleats a lot!. I use the bike for commuting and road-ish training. For commuting I miss the thing I mentioned in the previous paragraph. For training what happens is that I'm used to pedal in a full circle, pulling towards front in the upper part of the circle and pushing rearwards at the bottom. Without retention, when pushing forward my feet tend to roll the pedal and fall towards the front of it. (Pulling rearwards is not problematic).
An alternative to straps or cleats
A friend of mine was not fond of cleats, but he was also afraid of complete pedal strap setup, so he got a set of straps that came with plastic toe clip and shortened it by cutting and re-joining with bolts and nuts, and removed the part of the clip where the straps are bucled so the left only a rather small hook that retained the toes and kept his feet ball-over-axle. (He did't used the straps).
He implemented this system on XC trails on a Trek VRX, and I could say it was very effective. The feet where easy to remove from the pedals, almost any kind of shoe could be used, It didn't add much wheigt. It surely helped to push forward on the upper part of the cycle.
After a while getting re-accustomed to my platform pedals, my commuting has become more fluent. My feet has learnt a weird to describe motion that allows me to move the pedal upward to have it ready to start the next sprint. However, I still roll the pedal over when training and trying to do an outburst...
The reason for straps is that properly tightened straps give you the best absolute power transmission, because you can both push and pull on the pedals at the same time, with zero risk of the pedals unclipping themselves. Clipless pedals, while they have been getting better, could historically unclip randomly at crucial moments such as sprints, with usually catastrophic results.
In the old times, you would find straps (toeclips) on road and track racing bikes, or bikes of people who had some pretence to racing. The advent of mountain biking occurred simultaneously to the invention of clipless pedals, so it's rare to find clips on the MTB. They are still used in track racing.
The other answers rightly point out that being able to pull up on peddles can help improve power output but completely overlook low speed bike control.
To help with balance when the bike is stationary it is very useful to be able to pull the forward pedal up just slightly and apply a tiny amount of pressure. This can allow you to stay clipped in and immediately accelerate rapidly when a traffic light turns green (because proper cyclists stop at red lights). And that applies to a fixie or a freewheel.
Toe straps and cleats both allow this kind of pedal control.
It helps those who have a disability that causes the foot to come off the pedal or keeping it in the safe natural position. I know its not safe for me to ride my mountain bike without it since my stroke last summer. It is a device created to help those with physical disabilities to be able to ride a bike.
I prefer using straps, been using them for 40+ years now and love 'em. As mentioned by previous posters pulling up can have its advantages. While setting myself up for a sprint I worked on bouncing the back wheel as I was already in motion, this has to be timed just right,the wheel would hop and you get that xtra spin like a burnout as you're just about to give it your all.
I once did that and my chain decided to break at the same time I was over the handlebars going for the finish, that hurt a lot skating on my chest on hot asphalt.
My one and only reason: Fixed / Brakeless (wearing flip-flops) Hence the best answer to the OP's question of why PEDAL STRAPS - (not why cages / clips are better/not better....sheesh)
To elaborate: My toesies get crunched in the cage and there aint no clips on the bottom of my slaps.
*btw, I love the "pull up pedal" answers.
I still use straps, i am the only one in the Gatineau hills who still uses them i think, but first of all people spend thousands on light bikes and then have a relatively heavy toe clip and shoe and pedal. Toe straps are a lot lighter. Also the expense. Bicycling is kind of a fetish of style and very snobby, i am an old bicyclist from the seventies. Finally did get bike shorts on sale though. Good upgrade, i am not sold on toes clips, i prefer straps, for weight, for lack of expense and for not joining the bike tech cult.