Just to elaborate on Qiaochu's answer: one way to prove quadratic reciprocity is to observe that, for an odd prime $p$, the quadratic field $\mathbb Q(\sqrt{\pm p})$ (the sign being chosen so that $\pm p \equiv 1 \bmod 4$) is contained in $\mathbb Q(\zeta_p)$ (the field obtained by adjoining a primitive $p$th root of unity to $\mathbb Q$), as can be seen by using Gauss sums, and combining this with the irreducibility of the $p$th cyclotomic polynomial (which shows that $Gal(\mathbb Q(\zeta_p)/\mathbb Q) = (\mathbb Z/p)^{\times}$).
All these concepts go back to Gauss's Disquitiones Arithmeticae, which served to ispire and guide all the subsequent developments in number theory in the 19th century.
Gauss himself introduced his Gaussian integers (i.e. the ring $\mathbb Z[i]$) as part of his investigations of biquadratic (i.e. fourth power) reciprocity. His student Eisenstein investigated cubic reciprocity (and introduced the ring $\mathbb Z[\zeta_3]$ as a tool to this end).
Later in the 19th century Kummer investigated higher prime power reciprocity laws, and was led to the invention of the main concepts of algebraic number theory (ideals, unique factorization into prime ideas, the class group and class number, all in the context of the fields $\mathbb Q(\zeta_p)$) as part of his investigation.
The investigation of higher reciprocity laws continued. When the class group of $\mathbb Q(\zeta_p)$ is non-trivial, especially when it has order divisible by $p$, new phenomena emerged, which led Hilbert to the concept of Hilbert class field.
Out of all this the general conception of class field theory emerged, and was finally established by Takagi in the early 20th century.
Hecke was aware of all this tradition, and it to this tradition and these developments that he is referring in his remark.