Short answer? Batteries.

Long answer. Batteries. Lots and lots of batteries.

Let's back up a little bit first. Instead of getting into details about resistance and current, line leakage and such, suffice it to say, the longer the line, the more juice you need to get a message from point A to point B. Along the line, many things could weaken the signal. Poorly designed, dirty or damaged insulators, poorly grounded equipment or wires, damaged wire, weak batteries, and more.

In order for the telegraph to function properly, the line had to be in good mechanical order; properly grounded and insulated, and fitted with the right sort of equipment. At the heart of it all were two different sorts of batteries; the Grove Cell, which had a delightful tendency to release toxic fumes, and the Daniell or gravity cell, which became the standard battery by virtue of not venting lots of gas, and being a safer, more efficient battery.

Banks of cells (batteries of cells if you will) would be installed at each station along the line (several hundred miles apart on the transcontinental line for instance) while on shorter lines, stations and batteries might be closer together, depending on the population of the area. You can find an interesting discussion on the physics of the first transcontinental telegraph line here and a good discussion of batteries and some other telegraph tech here (The author is clueless about telegraph insulators though, ignore that bit).

So in essence, it was large banks of batteries situated at regular intervals along the line, as was required by the operating needs of the given line.