If a liquid such as whisky or water is distilled, it is heated until it changes into steam or vapour and then cooled until it becomes liquid again. This is usually done in order to make it pure.
If an oil or liquid is distilled from a plant, it is produced by a process which extracts the most essential part of the plant. To distil a plant means to produce an oil or liquid from it by this process.
If a thought or idea is distilled from previous thoughts, ideas, or experiences, it comes from them. If it is distilled into something, it becomes part of that thing.