I think this question has no easy answer but I'll give it a shot anyway (beware: oversimplification ahead!).
The main idea of the Malliavin calculus is to be able to differentiate stochastic processes like Brownian motion (or more general martingales with bounded quadratic variation), which are not differentiable in the traditional sense (because of their infinite variation).
Insofar the Malliavin calculus is the natural counterpart for stochastic differentiation to what the Ito calculus is for stochastic integration.
One of the practical application of Malliavin calculus is in the area of calculating option Greeks which makes sense since you would suspect that you needed derivatives to calculate these.
The main problem with the traditional approach is that the derivative needs to be approximated using the ﬁnite diﬀerence method and such approximations can become very rough. The integration by parts formula obtained from Malliavin calculus can transform a derivative into a weighted integral of random variables. This gives a more accurate and fast converging numerical solution than obtained by the classical method.
Some parts of the following thesis (on which parts of this answer are based too) might be helpful to dive deeper into the matter: The Malliavin calculus by Han Zhang.
To dive deeper into the practical applications (plus a primer on Malliavin calculus at the end!) can be found here: Smart Monte Carlo: Various Tricks Using Malliavin Calculus by Eric Benhamou.