Straight to the battery? that'd be your issue, instead of going through a fuse to lower the amperage on the 12V it gave every amp in the battery through that small wire, thats why it's always important to fuse things coming off a battery.
Not quite true. While fusing them is a good idea if you are running them directly off the battery, fuses don't do anything to lower the current flowing
unless the current exceeds the fuses rating. In that case it will just blow, and thus you will have 0 current.
Also, a car battery is only 12V
with the car not running. When the engine is on and the alternator is charging the battery, the voltage will be closer to 13.5-14V. Sensitive electronics don't like changes in voltage too much. LED's are current driven devices; this means that the amount of current directly dictates how much light they will emit (ie their brightness). One characteristic of this is that a small change in voltage results in a large change in current. Thus for a little extra voltage, the LED will get A LOT brighter, and also must absorb more power and produce more heat. Prolonged exposure to high currents (and subsequently high heat) will drastically shorten LED lifespan. In other words, apply as close to the required voltage as possible.
Since this is a strip, it will be pre-fabricated with current limiting resistors to allow the LEDs to survive 12V and last a long time. Unless you got some nice high dollar ones with a built in constant current source......but thats for another day
On to your problem of them staying on all the time. One way to do this is to get a rocker switch and put it somewhere easily accessible. Just wire it to the load side of the fuse (if you pull the fuse out of the socket, take a voltmeter to the 2 sockets. The side without voltage will be the load side that you will want to connect to.) and then from there put a switch in series with the LED strip to cut it on and off. Wire the other side to ground.
Also, you could just tap into the interior lighting wire and have the LEDs act just like the other interior lights.
[+]---------*~*---------/ -------|>|--------[-]
Positive.......Fuse........Switch.......LED.........Ground