As stated above, the particular details depend on the policy, but from my experience with CSA:
How exactly do they work? Supposing I have to cancel my trip a week
prior to departure. Can I just say "I'm not traveling" and expect to
be covered by the policy?
Yes. The catches are:
The policy will only cover expenses which are "forfeited and non-refundable". This means for a typical non-refundable airline tickets the only cost which is forfeited is a change fee. So for a $1000 ticket you might only get refund of $300, which is a change fee, as the rest of the ticket is not forfeited. For hotels you'll only get reimbursed of non-cancellable reservations, and the rest you'd be canceling yourself.
The policy typically doesn't reimburse 100% - most of them reimburse 50-90%; CSA reimburses 75%. So in essence you'd only recover some costs.
Cancellation for no reasons is typically limited to 48-72 hours before departure.
Does the insurance cover for the reservations made apart from the
flight tickets? (Such as match tickets, rail tickets within the
destination country, accommodation, etc which are made by self and not
through a tour operator)
Yes, as long as they are not refundable, forfeited and included into total covered amount. For example, flexible rail tickets could often be canceled for free or with small penalty; in this case the insurance would only reimburse the penalty.