One way I know to get such information is by using PowerShell in the Package Manager Console, from within Visual Studio.
The Package Manager Console is a PowerShell console within Visual
Studio used to interact with NuGet and automate Visual Studio.
Basically you can use the Get-Package cmdlet to get a list of packages referenced in a specific project (or in an entire Solution). Regarding the license information for each package, for what I've seen you can only get the license URL and not just a short string representing the license type.
Here's an example for a Solution of mine returning a list of entries, each one consisting of the package identifier and the link to the license:
Get-Package | Select-Object Id,LicenseUrl
The output is something like this:
Other elements that can be returned are documented in the Nuspec reference, in the metadata section (e.g. the version of the package, a short description, etc.).