If you pursue a technical masters, say in ME or EE, you will learn a lot of the theory behind the plug-and-chug equations that you plow through say junior year (fluids for ChE, lots of canned equations that you had to plug in appropriate values/units from basically the equivalent of a word problem). In a graduate fluids class, you look at the fluid contours along the pipe wall or aircraft wing or study turbulence, which then explains what all those terms in the plug-and-chug equations really mean.
So if you are technically inclined, a bit academically focused, and go through the masters program either full-time or employer paid at night (terribly time consuming and invasive if you have say a family), you can then use these skills to do difficult designs or figure out new algorithms or understand test results or do difficult analysis or understand controllers or whatever.
You can also get a masters in systems engineering or project management or an MBA, where you are learning how to manage a team or how to combine 6 specialty inputs into a single product or whatever. You are then more suited to a management or systems role.
Or , with a bachelors you can use superior people skills or marketing skills or get it done skills or learn on the job or pick the brains of 10 experts in your organization and be really successful, possibly even doing really difficult technical work (although it may be hard to get assigned this over the obvious choice of person with the masters degree).
You can also change employers or even fields to move less technical (some people never want to see a textbook again after 4 years) or more technical (subject matter expert or inventor or whatever) or towards management.
The easiest path to more technical work is likely a technical masters that your employer or future employer finds related to the work they do, the easiest path to management coming from outside an organization is likely a technical management or systems type degree … but don’t discount high-achievers with less education ending up being king of the hill either.
Also some experimental type work or prototype development is actually really technical requiring advanced coursework … and requires hands on mechanical skills and interests …
So there is no one path fits all. I would also argue that at most your degree affects starting and low-mid-level salaries, there are huge ranges of individuals with different talents, engagement in work, ability to please their management, good or bad fits to company culture, etc that really affect whether you are making 100K or 150K or 200K at age 50. I don’t know if putting off employment for 2 years and earning 0 really will be offset with a 20K jump in starting salary … again lifetime earnings will really vary wildly based on a lot more factors than education.
Some fields where scientists and engineers intermingle might require more education to be seen as a core team member or a technical person, if everyone has a PhD in physics, that PhD in engineering might help you get invited more into the inner circle …
For a junior in college, these discussions with professors are sort of the understand the graduate student reality but also to see … are you really more academically focused or not ?
Rule of thumb is that you get 2 years work experience credit for a masters, but salary and work years don’t really correlate that well out past 10 or 15 years since you are the product of how hard you work and how much you learn and how useful you are more and more as you continue working.
And some companies do want masters degrees for their managers or technical senior staff.
And education is becoming higher with time, way more PhDs now than in the past, but engineering is not really rife with MS vs BS degrees even now, even in technical fields.