There are multiple ways to integrate, from sending leads to Lead Assign first and populating Hubspot downstream, to grabbing opportunities from within your Hubspot account and routing to your network of salespeople.
Getting leads from Hubspot to your Lead Assign account platform generally requires the use of a Zapier account to act as a bridge. Use the API key on the Integrations page and use it as a web hook. Oauth is used to get into the Hubspot account from Zapier.
For more information regarding Zapier Integration with Lead Assign, go here. For more information regarding Zapier Integration with Hubspot, go here: https://zapier.com/apps/hubspot/integrations.
It is recommended to use either: 1) New Form Submission Triggers, when a form is submitted, or 2) New Contact in List Triggers, when a contact is added to the specified list. This will depend on when you want the lead to go through the Lead Assign platform.