I am an aspiring physicist and once, I asked my professor on what triggers quantum entanglement and he graciously remarked "The great uncertainty principle!" - I was slightly confused and didn't say anything but pondered about it quite heavily on my way to home. Can somebody explain this?
As far as I understand, it is not.
Entanglement is a consquence of the Hilbert space structure of composite systems in quantum mechanics. When you postulate that if you have system A (with Hilbert space $H_A$) and system B (with Hilbert sapce $H_B$) then the Hilbert space of both systems together (i.e., when they interact) is $H_A \bigotimes H_B$, then entanglement comes naturally.
I think that you can be a little far-fetched to give meaning to the claim of your professor, but, IMHO, this is a more natural way to see things.
The uncertaintity principle is a consequence of more fundamental principles and it doesn't trigger anything.
In particular, systems evolve according to evolution equations such as wave equations.
In particular measurements correspond to operators.
In particular (strong) measurement outcomes correspond to eigenvalues of the corresponding operators.
In particular during measurement the probability current splits waves into pieces whose relative size is proportional to the relative size of the projection of the original wave onto the various eigenspaces corresponding to the various eigenvalues. For large numbers of repeated results in identically prepared states thus leads almost surely to relative frequencies of observation that are very close to the relative size of the projections onto those eigenspaces.
All of that follows from the evolution equations and which explains the above. And the above explains absolutely every experimentally observed result.
So where is the uncertainty principle? Using the above you can find out how often you get each eigenvalue. From that frequency you can find the mean (average) and the squared deviation from the mean (the average of the square of the difference between each particular result and the average result). You can compute this deviation for any pair of observables and you can multiply them together.
The uncertainty principle merely describes a lower bound to these products. But we could already compute the products for any particular setup and get the actual product rather than a lower bound. The usefulness of the uncertaintity principle is in a universality, that sometimes you can get a lower bound that works for lots of states (lots of setups) maybe even all of them. And this can tell you something useful about all the possible setups.
But it doesn't cause anything. It doesn't trigger anything. The actual evolution of the wave causes things.
As for entanglement. Measurements themselves are a kind of entanglement, where you entangle the measurement equipment with the thing to be measured, obviously you want the state of the measurement device to become correlated to the state of thing being measured. And entanglements between particles are a kind of half measurement, where they've become correlated but haven't (yet) separated into different results because you haven't yet selected which operator to measure so you don't have different eigenspaces to split into.
So two entangled particles have started down the road to measure each other, thus when you finish the job on one of them it ends up finishing the job on the other one. What's fun about it is that the other one can be far away and also entangled can be passed around.
But all the questions can be answered by looking at the actual evolution for the actual setup. And the uncertaintity principle, at best, merely describes universal lower bounds to things you could already compute for any actual setup. It never causes anything.