Click here for the Subtitle & Logo Removal Guide
The easiest way is to crop them out, but it changes the size of your video (usually shorter/wider). If it's 4:3 (1.33:1) footage then you can say you wanted to make it more cinematic looking and cut out enough to make it 16:9 (1.78:1) widescreen.
But if you are cropping 16:9 footage then you'll end up with even shorter or wider video.
This is why the general recommendation is to use DVDs as source since practically all of them have switchable subs and when ripped off the disc you get clean animation. Not to mention the generally better picture quality.
If you have
to use downloaded anime (for new shows just airing) then either try to find raws or videos with soft subs. You have to find these videos on your own if you go this route (Forum Rules
If the above options aren't satisfactory, you could always do it the hard way which is to edit each frame in Photoshop by hand.