It's not 'cut down', just 'cut' or often 'cut out'. It basically means taking one, apparently unecessary piece out of a process to make it more efficient or direct. It usually refers to selling things. For example, imagine a furniture factory makes furniture, sells it to furniture stores who then sell it to people. The factory could just sell furniture directly to people instead of the stores, cutting out the middleman, i.e. the stores. In the article you mention, the government doesn't normally build private schools (or they wouldn't be private). But it appears they have started doing so, cutting out the middleman. It doesn't say who exactly the middleman was--apparently in the past some other organization built private schools.

The expression "to cut out the middleman" means to eliminate intermediaries or go directly to the source in a transaction or process. By doing so, one often aims to reduce costs, increase efficiency, or gain more control over the outcome.

Usually it's 'to cut out the middleman.' A middleman is someone who is in the middle/intermediate to a transaction. It usually implies some kind of inefficiency or added cost as the middleman either takes a cut of the financial transaction or otherwise can slow down the transaction. In this case, the government is giving money directly to the school rather than giving the money to an intermediary/middleman who then dispenses the money to the schools.