Not really much to explain. This is a laborious process of creating selections or painting directly on transparent layers, then adjusting opacity and blending modes to colorize the original photo. Depending on teh type of image and in particular the color-based contrast in it, simple techniques like applying a gradient map adjustment can also be used to do part of the work.
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This is a very laborious process with Photoshop and in the professional world there are better alternatives. I work in advertising and we often use colorization as the effect, although never realistic is very good at attracting the human eye, and that is what visual advertising is all about. We use several software products for this and some of it will be outside the price range of the average user here, but a good package comes from Blackmagic Software. The software uses something called Neural Algorithms which uses artificial intelligence to recognise things like skin and clothing and so avoids the necessity of making manually precise selections. The results can be very good without the user being a fantastic artist. Prices range from $49 up to $200 depending on the features required.