Coincidences, like magic, can fulfil two roles.
- Cause problems
- Solve problems
Coincidences that make problems
To cause problems, the audience needs no explanation. After all, we read stories to see characters deal with unexpected problems (that's pretty much all plots).
How many movies, TV episodes, books, etc. have a plot that goes "there once was a person and something-or-the-other just so happened to happen right by them and, as a result, plot..."?
Four children just so happened to stay in a house with a magic portal hidden in a wardrobe and just so happened to find it.
Some guy just so happened to get bitten by a spider that just so happened to have been changed in some way and it just so happened to give him spider-like powers.
Some young man just so happened to find a gap in the fence between the normal and magic world and ended up going on an adventure (and finding love).
Watch or read Dirk Gently (very different despite the names). These stories pretty much run on this but turned up to eleven.
Coincidences that solve problems
To solve problems and have it be a satisfying read is a whole other story. Brandon Sanderson's first law of magic goes something like, "The ability of magic to resolve the plot in a satisfying way is directly proportional to how well the audience understands the magic".
If I were to create a cheesy ripoff, Matthew Brown's first law of coincidence would state, "The ability of coincidence to resolve the plot in a satisfying way is directly proportional to how far in advance the audience sees the coincidence coming". That's pretty much how dramatic irony works.
In other words, if you drop new information in right at the end, readers will feel cheated; if, on the other hand, you telegraph this information, allow the characters to discover it in the nick of time - especially if the readers are screaming at the characters to notice - and then they can act on this information, it will still be satisfying to read.
These examples carry mild unmarked spoilers.
For example, Luke Skywalker just so happened to be the son of Darth Vader.
For example, Thr3e by Ted Dekker. (The book is good, the film stinks IMHO). The protagonist keeps running into apparent coincidences that connect him to the antagonist. By the reveal (at least, in the book) everything comes together and it becomes clear that this has been telegraphed from the start.
For example, Harry Potter just so happened to have the matching wand to the big bad (which saves him).