Whether API Management can do this, depends on what you exactly mean by these requirements.
Route traffic to various microservices As you mention the microservices are Azure Web Apps, I assume you mean different microservices to route to, based on a different endpoint. You can route a request based on the contents to a certain endpoint.
Cope with traffic demands and scaling Azure Web Apps are scalable by default, and the traffic manager takes care of it. APIM can only handle traffic demands and scaling to the platform itself. You can scale up or out, even automated based on rules. However, as scaling might take some time it's recommended to monitor the capacity metric to accomodate for increasing load.
Support API versioning APIM is 'just' an virtualization layer between the customer and the API. So having API versioning on APIM only makes sense when you actually do versioning on the API. In APIM you can create version sets which specify the versioning strategy for the API, based on header, path or querystring. An API can be deployed in APIM based on the version set, which makes it a versioned API.
