I cannot understand following wikipedia illustration:
It shows the antenna at right angles to the direction of travel of the EM wave. The wavelength is measured in the direction of travel. So why does the relative length of antenna to wavelength matter?
Suppose you have a receiving, dipole, antenna. Ignore the existence of free space around the antenna — ignore that wavelength — and just think about the portion of the electromagnetic field immediately around the antenna. The field exerts a force on the electrons in the antenna, perpendicular to the propagation of the wave (or more precisely, in the same direction as the polarization of the wave). That is where the right angle comes from.
The wavelength of the wave in space is irrelevant, so far, because the antenna elements don't see it, they just see a locally oscillating field.
Now think about what happens in the antenna conductor. There is a force causing the electrons to move (a current). On the other hand, the antenna has ends, and current cannot flow out the end of a wire (outside of conditions that do not apply here).
Consider just the field impinging on some electrons near the middle of the antenna and ignore the rest of the field. They can start to move, and just like any other change in current in a conductor, it propagates, as a wave, along the conductor at a speed close to (but not equal to) the speed of light. When this change reaches one end of the wire, current can no longer flow there, so just like any wave hitting an obstacle it reflects back and reaches its starting point, and there are standing waves within the antenna.
You've indicated that you already understand the idea of sound waves and standing waves in a pipe, so I'll skip going into more detail there. Just note that the proper analogy purely from the perspective of analyzing standing waves is:
The there is no obvious direct analogy for the interaction of the EM wave since it is spread out along the entire length — it's like you have a series of fans in the pipe, not like an outside pressure wave passing through an opening.
To summarize: the two lengths are similar not because the extent of the wave in free space maps somehow to the extent of the wire despite being at right angles, but rather because there are two wave phenomena of the same frequency and almost the same propagation speed.
So why does the relative length of antenna to wavelength matter?
A monopole antenna (for instance) can be "short" and it will pick up a signal that is proportionately smaller AND look like an impedance that is highly capacitive to the receiver. The "resistive" part of the signal will also be very small too: -
Picture taken from here
This is a good thing for crystal radios because at (say) a length of 0.05\$\lambda\$ it can reactively tune with a coil and produce a decent Q factor in order to give good selectivity in the crystal radio.
On the other hand, for a transmitting antenna, this is problematic because you have to do two things: -
So, transmitting antennas are chosen to have a length that makes the electrical interface simpler. For instance, at 0.25\$\lambda\$ the impedance is purely resistive at about 37 ohms. You could even choose a length that is a bit short of 0.5\$\lambda\$ and get a resistance of over 2000 ohms with no reactive part.
If you go to bigger antenna lengths then you get a repeated pattern: -
Picture taken from here
The base of the graph is in MHz with a quarter wave being at 2.5 MHz. The reactive part of the impedance is blue and the resistive is red with both being in ohms along the y-axis. There are some discrepancies in the amplitudes between the two pictures but this isn't the point - the point is that antenna length affects impedance greatly and it steps and repeats as you go from an electrically short antenna to an electrically long antenna.
Regarding the antenna pattern, a dipole looks like this with the antenna vertical and at the centre: -
Picture taken from here
