Will trailer be my best option? If not, what is? I need to take a  backpack with me that's literally wider than my body and about as tall as my torso and neck, it weighs around 15-20kg.

What I'm concerned about is roads that aren't so good or places where I might even need to go offroad (I don't know) as well as be essentially having less mobility and being wider. 

I'm going to travel (mostly on my bicycle) through various countries, which includes central European countries as well as northern countries, including their northern parts (so northern Sweden for example).

I'm low on budget though, keep in mind, so it isn't going to be something super fancy. Do you think a trailer will be my best option, and if not, what kind of attachment is?
From your other question - your budget for the bike was 300 euros.  You'd spend that much on a trailer alone.

Trailers more than double your rolling resistance, and increase your windage area a lot.  Riding long distance with a 2 wheel trailer is unpleasant.

You might build your own trailer, there are many plans around the web for this, using a square of metal or timber, two wheels and either a seatpost or a rear-axle hitch.

Another possibility is a single wheel trailer, which carry a bit less but are more useable on singletracks and narrower roads.

Making a trailer with a locking bin could solve your security problem, but it still needs locking closed and locking up to something when unattended.

In order of preference, top of the list would be front and rear panniers, then a frame bag, then a trailer, then a rear rack with your pack on it, and lowest on the list is a backpack on your back.