Why is 7 TeV considered as a big amount of energy?

$7 mathrm{TeV}$ is not that much kinetic energy, that has been covered by your question and previous answers.

However, in the context of a proton, with a rest mass of $1.672times10^{−27}~mathrm {kg}$ (very, very little mass), when a single proton has $7 mathrm{TeV}$ then it is traveling at a specific speed:

$$E= mc^2$$ begin{align}E& = E_0 + E_mathrm k E_mathrm k&=E- E_0 &= mc^2 -m_0c^2 &= gamma m_0c^2 - m_0c^2 &= m_0c^2 left(frac{1}{sqrt{1-frac{v^2}{c^2}}}- 1right) implies 1+ frac{E_mathrm k}{m_0c^2}&= frac{1}{sqrt{1-frac{v^2}{c^2}}} implies {sqrt{1-frac{v^2}{c^2}}}&=frac{1}{1 + frac{E_mathrm k}{m_0c^2}} implies 1-frac{v^2}{c^2} &= left(frac{1}{1 + frac{E_mathrm k}{m_0c^2}}right)^2 implies frac{v}{c}&=sqrt{1 -left(frac{1}{1 + frac{E_mathrm k}{m_0c^2}}right)^2 } end{align}

For a proton at $7 mathrm{TeV}$, this is $99.9999991 %$ times the speed of light

Source

Now, keep in mind this is for each proton in two beams, each proton having $7 mathrm{TeV}$, traveling through a superconductor cooled by helium, and then colliding for a sum of $14 mathrm{TeV}$ in each proton-proton collision.

Each beam contains $2,808$ 'bunches' of protons, and each 'bunch' contains $1.15 times 10^{11}$ protons, so each beam then consists of $362 mathrm{MJ}$ (megajoules).

This gives the total kinetic energy of $724 mathrm{MJ}$ just in the beams alone: about 7 times the kinetic energy as landing a 55-tonne aircraft at typical landing speed according to the Joules Orders of Magnitude Wikipedia page, or with that $59~mathrm{ m/s}$ landing speed, 218.9 tonnes, so about a safely loaded Airbus A330-200 (maximum takeoff weight of 242 tonnes)

Add to that the energy required to keep the ring supercooled enough so it remains superconductive, accelerate the beam in the first place, keep accelerating it so it doesn't lose velocity, light and heat and power the facility.

"At peak consumption, usually from May to mid-December, CERN uses about 200 megawatts of power, which is about a third of the amount of energy used to feed the nearby city of Geneva in Switzerland. The Large Hadron Collider (LHC) runs during this period of the year, using the power to accelerate protons to nearly the speed of light. CERN's power consumption falls to about 80 megawatts during the winter months."

- Powering CERN