안녕하세요:) David 님!
(감기에 걸리다)는 to have a cold라고 생각하면 돼요.
This is an expression to say To catch/have a cold.
and the verb 걸리다 has so many different meanings
but for that one, it means to fall ill (to catch)
and this verb 걸리다 doesn't come with an object particle which is 을 or 를.
Instead, it is usually used with -에 (postposition)
Hope this helps^^