Abstractive summarization at controllable lengths is a challenging task in natural language processing. It is even more challenging for domains where limited training data is available or scenarios in which the length of the summary is not known beforehand. At the same time, when it comes to trusting machine-generated summaries, explaining how a summary was constructed in human-understandable terms may be critical. We propose Multi-level Summarizer (MLS), a supervised method to construct abstractive summaries of a text document at controllable lengths. The key enabler of our method is an interpretable multi-headed attention mechanism that computes attention distribution over an input document using an array of timestep independent semantic kernels. Each kernel optimizes a human-interpretable syntactic or semantic property. Exhaustive experiments on two low-resource datasets in the English language show that MLS outperforms strong baselines by up to 14.70% in the METEOR score. Human evaluation of the summaries also suggests that they capture the key concepts of the document at various length-budgets.