It means "like, in the manner of". E.g. He was dressed à la cowboy. (He was dressed like a cowboy)

In truth, "à la" is only really used sardonically as it's quite a pretentious turn of phrase in English. It comes from French, as you say, and was popular on restaurant menus when they used to be written all in French. However, you can use it if you want to, especially in the constructions 'à la mode' (in fashion), à la carte (a type of service in a restaurant) along with the Italian phrases 'al fresco' (outdoors) and 'al dente' (crunchy), both of which are understood as well. 'He made me an offer I couldn't refuse, à la Godfather' ("along the lines of" is probably the best English translation for it)