Whoa, really? Their answer was pretty spot on to your question. Why take such a hostile attitude when it’s your own inability to understand? I’m shocked to see such an ignorant and condescending comment being upvoted.
There’s nothing wrong with not understanding a technical explanation and asking for it to be simplified to your level of understanding, but this is not a simple concept to explain because it is ACTUALLY complex and counterintuitive. How is it their fault?
I'm just trying to answer your question, but I don't think you're exactly clear on what you want to know. It's ok, no need to get defensive about it.
Whether or not 'one end of the wire gets heavier' depends on what your doing. If you are using the wire to power an LED from a battery, then no, because electrons are removed from one end and placed into the other at equal rates. Charge and mass within the wire are both globally and locally conserved.
If you do something where charge/mass isn't conserved such as removing electrons from one side (i.e. by rubbing a fork on a carpet) or by using an electric field to 'tilt' the electrons to one side, you can create a (very small) mass imbalance. This activities are not usually considered to be useful electrical current.
