The diagram shows the construction of the system. Note that this is designed for a single band only. The heart is a coax switch. Tee connectors A and C are screwed directly on the coax switch ports. It is important to use a coax switch that does not ground the unused positions. Quarter wave lengths of coax provide impedance matching.
It is important to realize that the physical length of a ¼ λ piece of coax is shorter than ¼ λ in free space. Typically it will be 60-90% of the free space length. You can look up the specs for the coax you have and calculate the proper length for the band you will use it on, but a better idea would be to start with lengths a bit on the long side and trim them to the proper length with an antenna analyzer.
The coax to the antennas (L1 & L2) must be the same length if this is to be used with stacked Yagis. They can be any convenient lengths if the antennas are of different types. Similarly, the coax from the switch to the transmitter can be of any length.
The system works because of the fact that lengths of feed lines act as impedance transformers. The transform function depends on the impedance of the feed line relative to the load, and the length of the feed line.
Two special cases are at work here. A ¼ λ feed line will convert an open circuit on one end to a short circuit at the other. The inverse is true, and a short circuit on one end will appear as an open circuit at the other end. The other special case is that a ¼ λ 70 Ω coax will transform 50 Ω on one end to 100 Ω on the other.
Looking at the diagram, the switch is in the center position. In this position, both antennas will be connected. Antenna 1 will present a 50 Ω load at Tee connector A. The 70 Ω coax between connectors A and B will transform the 50 Ω to 100 Ω at connector B. Similarly, the impedance from Antenna 2 will also appear as 100 Ω at connector B. The two 100 Ω impedances in parallel form 50 Ω which is preserved through the 50 Ω coax back to the switch and to the transmitter.