The Azure API Management service doesn't have a free tier you can use to try and test it. Even the Developer tier has a monthly cost (you can use the Consumption tier, which allows you to have a number of free calls each month, but, with that tier, you don't have the developer portal).
So, the question may be: Why should you use it and pay?
One of the answers may be that it depends: it depends on what your business is, what kind of API you have to expose, how much chance you have of modifying your APIs, and so on.
Of course, API Management is not the answer to every scenario, so you should evaluate carefully whether you should use it.
In this section, we'll try to list all the advantages you can leverage if you use API Management, while the bigger disadvantage is, of course, the cost.