Your calculations are correct in essence. For a 1440p60Hz signal, you have a data rate of 5.8Gbps once you allow for blanking time as well (non-visible pixel border in the image output).
For HDMI/DVI, a 10/8b encoding is used, which means effectively although you have say 24bit of colour data per pixel, actually 30bit is sent as the data is encoded and protocol control words added. No compression is done at all, the raw data is sent, so that means you need 7.25Gbps of data bandwidth.
Again looking at HDMI/DVI. It uses the "TDMS" signalling standard for data transfer. The HDMI V1.2 standard mandates a maximum of 4.9Gbps for a Single-Link (3 serial data lines + 1 clock line), or in the case of Dual-Link DVI a maximum of 9.8Gbps (6 serial data lines, I think). So there is more than sufficient bandwidth to do 1440p60 through a Dual-Link DVI, but not through a HDMI V1.2.
In the HDMI V1.3 standard (most devices actually skipped to V1.4a which is the same bandwidth as 1.3), the bandwidth was doubled to around 10Gbps which would support 1440p60, and is also enough bandwidth for UHD at 30Hz (2160p30).
DisplayPort as another example has 4 serial data streams, each capable (in V1.1) of 2.16Gbps per stream (accounting for encoding), so with a V1.1 link you could do 1440p60 easily with all 4 streams. They have also release a newer standard, V1.2 which doubles that to 4.32Gbps/stream allowing for UHD @ 60Hz. There is a newer version still which they have pushed even further to 6.4Gbps/stream.
Initially those figures sound huge, but actually not so much when you consider USB 3.0. That was released with a data rate of 5Gbps over just a single cable (actually two, one for TX, one for RX, but I digress). PCIe which is what your graphics card uses internally nowadays runs at up to 8Gbps through a single differential pair, so it is not all that surprising that external data interfaces are catching up.
But the question remains, how is it done? When you think about VGA, that is comprised of single wires for R, G, and B data which are sent in an analogue format. Analogue as we know is highly susceptible to noise, and the throughput of DAC/ADCs is also limited, so that massively limits what you can push through them (having said that you can barely do 1440p60Hz over VGA if you are lucky).
However with modern standards we use digital standards which are much more immune to noise (you only need to distinguish high or low rather than every value in between), and also you remove the need for conversion between analogue and digital.
Furthermore the advent of using differential standards over single ended helps significantly because you are now comparing the value between two wires (+ve difference = 1, -ve difference = 0) rather than comparing a single wire with some threshold. This means that attenuation is less of an issue because it affects both wires equally and attenuates down to the mid-point voltage - the "eye" (voltage difference) gets smaller, but you can still tell whether it is +ve or -ve even if it is only 100mV or less. Single ended signals once the signal attenuates it might drop below your threshold and become indistinguishable even if it still has 1V or larger amplitude.
By using a serial link over a parallel one, we also can go to faster data rates because skew ceases to be an issue. In a parallel bus, say 32bit wide, you need to perfectly match the length and propagation characteristics of 32 cables in order for the signals not to move out of phase from one another (skew). In a serial link you have only a single cable, so skew can't happen.
TL;DR The data is sent at the full bit-rate you calculated (several Gbps), with no compression. Modern signalling techniques of serialised digital links over differential pairs make this possible.