The short, very simplified version is: if you're a VAT registered business, you add VAT to any in-scope goods or services that you sell. Your accountants make a note of how much VAT you've received in this way. You then add up the total amount of VAT that you've paid to other companies for the goods and services that enabled you to sell what you did. You subtract this from the amount you received, and that's what you pay to HMRC.
Whether you were able to do this previously depends, again in very simplified terms, on whether the services you provide were zero rated (VAT is chargeable, but at 0%, and you can reclaim VAT on what went into it) or exempt from VAT (VAT is not assessed at all and you can't reclaim). I believe school fees used to be exempt entirely, but don't quote me on that.
If something changes from zero rated to standard rated, then the price to the consumer will very likely have to go up by the full 20%, because the supplier was already reclaiming the VAT on their costs. If something is brought into the scope of VAT that was previously excluded completely, then the associated costs of providing it essentially drop because it becomes possible to reclaim the VAT on expenses where it wasn't previously. In this case, if only the added costs are being passed to the consumer, it'll go up by 20% of the company's profit margin on those services, not 20% of the full price.