There's no one definite reason why my wife and I commute, but in the long run it is less expensive. To us it is fun, healthy, empowering, sometimes quicker, and yes it saves us on gas and maintenance on our cars.
I think the key is buying quality products/ accessories for your bicycles. I find that there are just way to many people on these forums looking for the least expensive options instead of quality products that last. There are a couple of old saying that come to mind. "The quality remains long after the price is forgotten" or "buy cheap and buy twice (or more)" and the good ole "pay me now or pay me later".
We use to spend at least $50.00 a week in gas between our two cars now we are down to about $10.00 a week which saves use $2080 a year just in gas alone. Plus we have definitely seen an improvement to our health, no more coughs and colds, blood pressure under control and generally just feeling better at the end of the day.
You mentioned fixing flats on your bicycle. Both the wife and I have logged over 2500 commuting miles so far this year, and we both had our first "puncture" within a week of each other. A $3.00 patch kit fixed both tubes. Our tires cost more up front but I feel that I am saving money in the long run when you consider how long the tires will last and the extra puncture protection they give. Our headlights and tail lights (absolute necessity for a commuter bike) also cost more than the bargain priced equivalents but we have had absolutely no problems or issues with them.
To us, it is cheaper in the long run, but we commute because it's fun and empowering