DL is a branch of methods/techniques in ML which itself is a branch of methods/technqiues for AI.
DL: Deep Learning is any machine learning method that involves the use of large neural networks. Its name refers to the fact that the neural networks often have many hidden layers deep within the system. "Deep learning" sounds sexier than "A neural network but with lots of layers".
ML: Machine Learning philosophically can be seen as an attempt to recreate artificial intelligence using data or practically as a branch of statistics and computer science that revolves around algorithms that compute data from which we can make inferences and predictions about response variables. Machine Learning falls entirely into both Computer Science and Statistics. It's called "Machine Learning" because it sounds sexier than Statistical Learning which was what it mainly went by before it became popular.
AI: Artificial intelligence means to recreate non-biological intelligence. Machine Learning (in research) is a group of different tools, techniques, methods and approaches to achieve this.
AI falls into two-categories: Weak AI (All AI we have now) and Strong AI (Human level intelligence). Over time researchers have used different methods to try to achieve this. Early on this was through logic and symbolic representation but in the modern world the most prominent is Machine Learning as a result of its recent successes.
It's important to bare in mind that not all of Machine Learning is specifically an attempt to recreate human intelligence or build strong AI, a lot of it is just to analyse data.