Fuel is needed to start a battle: to attack a convoy or another player's base. If you want to start campaign mission you need 10 fuel as well. If your base was destroyed and you'd like to revenge, you also need 10 fuel.
Fuel is restored automatically and reaches 80 units (if your Fuel Storage is maxed out). Full recovery takes 6 hours. Recovery process starts as soon as your fuel amount falls below its limit (for example, 79 units out of 80).
Aside from that you can receive fuel:
1) Purchasing it with credits, clicking on '+' next to it.
2) Asking your friends for assistance (5 requests is the limit, which equals 50 fuel).
3) Collecting it from destroyed mechs after an attack on your base.