Consider a decision problem. A decision can be either "yes" or "no". So. we can divide the problem cases to 2 parts- one where the answer is "yes", and one where the answer is "no".
Now, if the "yes" instance of the problem is given, suppose we (our TM) can always say "yes", then the problem is partially decidable (its language is recursively enumerable).
If we can also say "no" for all "no" instances of the problem, then the problem is decidable (its language is recursive).
NB: TM for a partially decidable but not decidable problem, will go to an infinite loop for SOME "no" instances. For some "no" instances it may say "no". (Obviously, it won't say "yes" for any no" instance)