Learn Hardcore Korean: jibeseo keompyuteo-e rogeu-inhaeyo.

In Korean, -에서 marks the place where an action happens. Since 로그인해요 (“log in”) is an action taking place at home, you use 집에서. By contrast, 집에 would mark a destination or a static location (“to/at home” without emphasizing an action).

Why is it 컴퓨터에 and not 컴퓨터를?

You “log in to” something, not “log” something. In Korean that target or endpoint of the action is marked with -에, so you say 컴퓨터에 로그인해요 (“I log in to the computer”). The object particle -를 is for things you directly act upon, which isn’t the case here.

Why don’t we say 로그인을 해요 with an object marker on 로그인?

로그인하다 is a compound 하다-verb (noun + 하다). When you conjugate it you drop 하다 and add 해요 directly to 로그인, giving 로그인해요. You don’t turn 로그인 into a separate noun object with -을/를 here.

Could I use 컴퓨터로 로그인해요 instead of 컴퓨터에 로그인해요?

No. -로 often means “by means of” or “toward,” but when talking about logging into a system or device, Korean consistently uses -에 to mark the target you connect to. 컴퓨터로 로그인 would sound like “I log in by using the computer,” which isn’t what you intend.