I put on an overcoat and went out so I wouldn't catch a cold.

Literal

Cold [object-を] catch bad-[so-といけないから] overcoat [object-を] wore-and went-out.

~といけないから means 'because it would be bad if' — a purpose/reason hybrid. オーバー is a loanword for 'overcoat' (from 'overcoat'). A practical precautionary action.