Why are basis sets needed?
Spatial orbitals $\phi_i$ in modern electronic structure calculations are indeed typically expressed as a linear combination of a finite number of basis functions $\chi_k$,
\begin{equation}
    \phi_i(1) = \sum\limits_{k=1}^{m} c_{ki} \chi_k(1) \, .
\end{equation}
In the early days, atomic orbitals were built out of basis functions, while molecular orbitals were built out of atomic orbitals, which is where the name of the approach, linear combination of atomic orbitals (LCAO) originated.
Today both atomic and molecular orbitals are built out of basis functions, and while basis functions for molecular calculations are still typically centered on atoms, they are usually differ from the exact atomic orbitals due to approximations and simplifications.
Besides, basis functions centered on bonds or lone pairs of electrons, or even plane waves, are also used as basis functions.
Nevertheless, the approach is still commonly referred to as linear combination of atomic orbitals. 



Now, to the very question of why do we do things this way?

On the one hand, the LCAO technique made its way into quantum chemistry as just another example of a widely used approach of reducing a complicated mathematical problem to the well-researched domain of linear algebra. To my knowledge, this was proposed first by Roothaan.1 However, and as it was mentioned by Roothaan from the beginning, the LCAO approach in present day quantum chemistry is also attractive from the general chemistry point of view: it is tempting to construct molecular orbitals in modern electronic structure theory from their atomic counterparts as it was done by Hund, Mulliken and others already in the early days of quantum theory2 and as it is though in high-school general chemistry courses today.



1) Roothaan, C.C.J. "New developments in molecular orbital theory." Reviews of modern physics 23.2 (1951): 69. DOI: 10.1103/RevModPhys.23.69

2) Pauling and Wilson in Introduction to Quantum Mechanics with Applications to Chemistry refer to the following works in that respect (p. 346):


  F. Hund, Z. f. Phys. 51, 759 (1928); 73, 1 (1931); etc.; R. S.
  Mulliken, Phys.  Rev. 32, 186, 761 (1928); 41, 49 (1932); etc.; M.
  Dunkel, Z. f. phys. Chem. B7, 81; 10, 434 (1930); E. Hückel ,Z.f.
  Phys. 60, 423 (1930); etc.


Hund's papers are unfortunately in German, but Mulliken's ones are quite an interesting read. Especially the second one,

Mulliken, Robert S. "Electronic structures of polyatomic molecules and valence. II. General considerations." Physical Review 41.1 (1932): 49. DOI: 10.1103/PhysRev.41.49

which (again, to my knowledge) introduced the very term "molecular orbital".