There are two aspects, scalability and inductive bias. DL is scalable because compositions of differentiable functions make backpropagation fast, and those functions being mostly matrix multiplications make GPU acceleration effective. Combine this with stochastic gradients, and you can train on very large datasets very quickly.
Inductive biases make DL effective in practice, not just in theory. While the universal approximation theorem guarantees that an architecture and weight-setting exist that approximate a given function, the bias of DL towards low-dimensional smooth manifolds reflects many real-world datasets, meaning that SGD will easily find a local optimum with these properties (and when it doesn't, for example on tabular data where discontinuities are common, DL performs worse than alternatives, even if with more data it would eventually approximate a discontinuity).