Bookshelf speakers can technically function just fine on their own, but the bass will typically be lacking due to the practical design limitations of a smaller cabinet. A subwoofer can also help bookshelf speakers perform and sound better by adding a second crossover and relieving them of needing to reproduce lower frequencies, thus allowing them to really focus on the mids and highs, which they’re generally optimized to do.
Most bookshelf speakers utilize a 2 way design with a single crossover between a small tweeter, which produces the highs, and a standard sized woofer (usually 5.25 or 6 inches). In general, the bigger a driver gets, the better job it does at reproducing lower frequency sound waves, i.e. bass, but that then detracts from its ability to produce the mid range waves. It’s an unavoidable trade off, a “jack of all trades, master of none” situation, so to speak. Even if a speaker is made big enough to accommodate a larger driver, it will get more expensive, heavier, harder to position and place, etc.
One solution is to utilize a 3 way design that sets a second crossover to a dedicated built in subwoofer, and there are plenty of these kinds of speakers, but they’re reservedly high end and usually very expensive. For those looking to stick to a low to mid range budget (which should be most people), you’ll almost always get the best bass and overall performance for your dollar with a pair of bookshelf speakers and a subwoofer over a higher-end pair of standalone speakers.
With a separate subwoofer, the driver and cabinet can be nice and big (most sub drivers are 10-12 inches), while the bookshelf speakers can remain compact, economical thus, and really focus on the mids and highs which a 2 way design that most of them employ is really meant to do.
Adding a subwoofer might seem intimidating, but it’s actually really easy because most of them are powered. All you need is an wall-outlet and an RCA slot on your receiver, which any decent receiver will almost certainly have. Many subs are wireless too, so you might not even need to hook it up to your receiver, actually. There are plenty of really solid subwoofers for under 200 dollars, and in tandem to that you can then focus on a pair of bookshelf speakers that are really powerful and performant, without worrying about the response floor and them needing to try to be two things at once.
To retirate: at non-high budgets, you’ll almost always get the most power and value with a pair of bookshelf speakers and a subwoofer as opposed to floorstanding tower or otherwise standalone speakers.