"I would love to find some more clear, simple and qualitative explanation of how Vlans do work."
For "how Vlans do work" are you asking about physically or logically? The former can be a bit of an unknown because proprietary techniques can provide the logical. Logical, though, might be explained, in context of what a LAN is/does as a VLAN is a Virtual LAN.
Before getting into Virtual LANs, what's a LAN? Well, LAN stands for local area network, which means a network where everything is "local", or near each other, possibly like a single floor of office building with lots of desks, each with a network node at each.
Consider early Ethernet, 10Base2 and 10Base5, each used a single cable running from node to node, in series. (One reason the network was "local".)
The other aspect of early LANs, all the nodes could directly communicate at L2. Logically, much multiple folk in the same "room" (in the [great] manner @Flavio Miranda described). LAN networks, could, and did, vary in size, but again like Flavio's description, large number of users (nodes) per room (LAN) (and/or how active/busy those people [nodes] are) does impact how well work can be conducted. I.e. number of nodes and/or activity level of nodes, directly impacts how well your network can work. To continue to work well, you often need to limit the number of nodes per LAN, but what do you do if you split your LANs and they still need to intercommunicate? The latter is provided by L3, but for our discussion, we will limit ourselves that a "LAN", is a LAN L2 segment, and not all the LAN L2 segments that might be found at one place.
The last is important. LAN might be used to refer to all the network components near each other, or it might be, and often is, used to refer to a L2 LAN segment. The latter allowing all nodes to intercommunicate, directly, using L2; Flavio's one room structure.
A VLAN is a virtual implementation of a LAN L2 segment. Ever been in a room that has large panels that can subdivide the room? Usually making two rooms where there was one, while, of course, allowing the two smaller rooms be restored to the original larger room?
That's what VLANs provide. They allow us to define multiple L2 segments on a single switch, just as if we might subdivide a room.
Much like subdividing a room, using panels, you loose some of the attributes of having a constructed room wall, but the trade-off in flexibility often makes using VLANs very attractive.
To make the distinction of LANs vs. VLANs, hopefully, clear, let's pretend I have business which has three sets of users, comprising 3 accountants, 5 engineers and 10 doing sales. Users in each group, need to share work across a network. No need, or desire, for anything beyond that. What might be your network design?
Would you perhaps suggest this business LAN (assuming all 18 folks are near each other, "local"), have 3 distinct LAN L2 segments, one for the accountants, one for the engineers and one for sales?
How would you "create" these 3 LAN L2 segments? Maybe a 4 port switch for the accountants; maybe two 4 port switches or one 8 port switch for the engineers; and, for sales, maybe a 4 port and 8 port switch, or two 8 port switches, or one 24 port switch, or some other combination.
All is great until the business hires two more accountants! Now what do we do? Well, we can replace the 4 port switch with an 8 port switch or we might connect another 4 port switch to the existing 4 port switch (BTW, the latter provides us 6 ports, as one port on each switch is used to interconnect them).
VLANs "mimic" real LANs (L2 segments). We might start with a single 24 port VLAN capable switch, and define on it 3 VLANs, one for the accountants, one for the engineers and one for sales, and assign the needed number of ports to each VLAN.
Or if the original switches were VLAN capable, when we need two more accountant ports, we connect the 4 port switch to another VLAN capable switch, with free ports, and assign some of those free ports to our accountants VLAN. I.e. just as we can extend a physical LAN L2 segment with additional switches, we can extend a logical/virtual (VLAN) L2 segment across multiple VLAN switches.
Hopefully, the prior helps you understand VLANs.
BTW, the value of VLANs, to L2 segments, is somewhat like slicing is to bread. ; )
