You can of course use analytics that will give you statistics, but will not answer "Why?". Gathering Customer Feedback is crucial and You have to be preprared for low response rates. For better understanding of Customers' interactions with Your site You can use tracking/heatmap solutions (i.e. Crazy Egg or Hotjar). But most important is to define goals and identify steps that lead to them - what do You expect Customer will do or act like? What needs to be done to acheive it? Then start monitor those steps, gather feedback, mae changes and see if this works looking at stats.
This is why you need goals for your website. Otherwise you don't know if it is working or not.
The goals should be about the website itself, not traffic to it. In other words, it can't be volume of visits etc., as that determines the success of your traffic generation efforts, not the website.
What are the actions people can take on your website? What do you want people to do? Most importantly: what is in it for them? What does a person get back for spending time on your website?
Track activities (the more visitor-centric the better) such as clicks on buttons, paper/info downloads, newsletter subscriptions, etc. The more you have to offer the client, the better tracking you will get.
You can see where people are dropping off from in Google Analytics. You can look at the overall conversion rate and see if it's effective. I don't recommend spending much before doing some best-practices oriented conversion rate optimization. I can help with that.
Aside from getting visitor feedback, usability testing can help catch a great many issues. When it comes to website usage, many visitors/customers might be able to tell you that something isn't right, but they may not be able to pinpoint what that "something" is. Usability analysts know what to look for and know how to spot pitfalls.
If I had to rephrase your question, you are trying to understand what part of the website were people able to understand, which parts were they not. Sure, analytics, heatmaps etc will help you a lot with it but nothing is better than actually talking to some people. This could be over chat where you could ask people if they understood what the service was about but is best done over call - I don't know your business to tell you how to get people on call but nothing beats the density and richness of information and surrounding context you can get when you get users on a call.
This is a large, important question with no single or simple answer.
Data and data analysis is key, as well as tracking what users are actually doing on your website. This can be done using Google Analytics and very inexpensive tools to see what users are actually doing, such a Crazy Egg. To do this correctly and effectively, requires having someone on your team with UX expertise who has done this kind of post-launch customer/user tracking and analysis and has acted upon the findings with measurable results.
Bottom line, the simple answer to your question is to hire a As someone who does this work every day, I will tell you that there are no magic answers. Do it well and your website does everything you and your customers/users want and expect it to do. Do it not so well will be forever confusing and you'll just be guessing and not knowing whether anything you are doing is doing anything.
My advice is to take the plunge and hire the right UX pro and realize the full potential of your investment.
Happy to discuss further!