Look at the right hand of that diagram. The part that says "Furnace". See, the beam of silver atoms (or potassium, as I did it as an undergrad) comes from heating up metal so hot it evaporates. Then, because you have a pretty good vacuum outside of the furnace and in the apparatus, the metal gas comes rushing out of the oven. The only reason it makes a beam is because of the hole in the side of the furnace. You could get a narrower beam by narrowing the hole, and making the channel it goes through before reaching vacuum longer, but then you're also going to be prone to clogging as the metal condenses after leaving the furnace. Even if you can avoid clogging, narrower aperture and longer pipe constrict the flow of atoms, making the beam less intense.
In short, the metal starts as a gas in a furnace that is going in every direction with a Maxwell-Boltzmann distribution in velocities, and you use some method to select only those atoms going in the direction you want, roughly. There will always be some spread in the velocities you can pick out, and practical concerns limit how narrow you can make that spread. If you turn off the magnet (or remove it) you'd see one beam with finite width (as you see on the left). Then the magnet with the field gradient operates on that beam that has both a spatial spread and a velocity spread to produce the two beams you identified as blobs on the right. The kind of mouth-like shape of the right hand pattern has to come from the details of the magnetic field the beam passes through. Obviously, the field gradient is not uniform in the $y$-direction.