The speed cube lube is the most important accessory for a speedcuber. When you buy a cube usually it is not lubed. And when a cube is dry, you use it a lot, the plastic inside is scratching. The cube starts turning worse and worse, and later it gets destroyed. But after you lube your cube it gets smoother and enjoyable to turn, but most importantly the plastic on the inside of the cube is saved from scratching.
The timer and the mat are pretty useful when you want to measure your achievements. Of course, you can use your phone or computer, but on the official WCA competitions, Speedstacks timer and mat are used and you should get a timer too, so you can get used to the feeling. There are more timers, not only the Speedstacks one (it's expensive) such as the QiYi Timer, the YJ Timer, and more. The mat is used to protect your cube from damage when you drop it.
The cube cover is used before you start a solve. In the official WCA competitions, cube covers are used, so before the competitor starts a solution they have 15 seconds of inspection, but before the time starts they can't look at the cube. You can also use a hard box (such as a cube one).