I found out that my ISP does deep packet inspection.
Can they see the contents of HTTPS connections?  Wouldn't having HTTPS ensure that they can't see the contents being transferred?

And can having a VPN protect me against deep 
packet inspection by ISPs?

Deep Packet Inspection, also known as complete packet inspection, simply means they are analyzing all of your traffic as opposed to just grabbing connection information such as what IP's you are connecting to, what port number, what protocol and possibly a few other details about the network connection. 

This is normally discussed in contrast to the gathering of NetFlow information which mainly collects the information listed above.

Deep packet inspection gives your provider a lot of information about your connections and habits of Internet usage. In some cases, the full content of things like SMTP e-mails will be captured.

HTTPS does encrypt the connections but your browser has to make DNS requests which are sent primarily via UDP so that data will be collected as will any unencrypted links or unencrypted cookies sent incorrectly without https. These additional bits which will be collected may be very telling about what type of content you are looking at.

The larger concern for most people is about data aggregation, by collecting this information a data scientist could create a fingerprint for your Internet usage and later associate with past activities or activities from other locations (when you are at work or are on vacation). Likewise, your service provider may choose to sell this to any number of organizations (possibly including criminal organizations) where it could then be used against you in ways. In many countries, people have an expectation that their communications are considered to be private and collecting this data very much goes against that privacy expectation. 

Another interesting aspect of this is in the cases like the US where this data may soon be sold it allows International communications sent to people, or servers, in the US be sold as well. Likewise, this could potentially allow every agency from local law-enforcement, military, tax authorities, immigration authorities, politicians, etc. a way to bypass long-standing laws which have prevented them from accessing this type of information, or important informational subsets within this data otherwise. 

A slightly different concern when this data can be sold is competitive intelligence / corporate espionage. In the scenario where a company does a lot of research-intensive work at their headquarters located in some small geographical location (think of pharmaceuticals or a defense contractor) selling that data makes it possible for anyone to buy all of the traffic from the local ISP where most of those researchers live and analyze what they are looking for when at home, possibly even directly from the ISP hosting the traffic for their corporate headquarters. If other countries aren't selling similar data it gives foreign companies and companies wise enough to try and buy this data a huge technical advantage. Likewise, it would also allow foreign governments to buy ISP traffic which includes the data from US (or other government) Officials homes.   

Imagine companies monitoring their employee's behavior at home or on their mobile devices.

This will likely have a chilling effect on activists and whistle-blowers as well.

Likewise, if credit cards or PII are sent in the clear to a poorly secured remote site your ISP's data set now has a potential PCI or PII regulatory issue on their hands. So this amplifies data-leakage problems of all types by making additional copies of the data leaked. 

With the examples I've just mentioned above, and there are hundreds of others, it should be easy to see why this type of data collection has a different level of importance to it than just metadata or basic connection information. Even if your ISP never sells this data they are collecting quite an interesting dataset. 

It's a security issue that definitely has a lot of potential long-term security implications. 
