Please help ... What is meant by 'contract' here in this context?

It's a fancy word for "buy". 

No native English-speaker would say that.  You don't "contract" an insurance.

It means to purchase or arrange. It should be "contract insurance." There is no need for "an,' but if "policy" were added after insurance, it would be correct.

Contract, is a legal term for making an agreement. A legal contract is enforceable in court. It should read:

     "Would you like to make a travel insurance contract"? If you want to use the word contract.

"Insurance" is an uncountable noun so you never say "an insurance." "Contract" is incorrect. It is not a synonym for "purchase." Would you say "I contracted a house" or "I contracted a loaf of bread"?

There is a name for "insurance for travel." It is "travel insurance." The sentence should be "Would you like to purchase travel insurance?"

just if you want to contact them

It means buy.  "Would you like to buy some insurance for your travel?".