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If you replace Ext by Tor, you are defining functions on the derived intersection of $Y$ and $Z$, let's call it $W=Y\times_X Z$ (this is the fiber product in the world of derived schemes, which by definition corresponds to this Tor).

One can then describe the (first) Ext group you asked about as almost functions on the derived intersection of Y and Z (for closed subvarieties). Let's write $i,j$ for the inclusions of $Y$ and $Z$ (yes I know $j$ is usually reserved for open immersions, but anyway..). Then the derived Hom (Ext complex) $Hom(i_*O_Y, j_* O_Z)$ can be calculated by Grothendieck duality for the proper map $i$ as $Hom(O_Y, i^!j_* O_Z)$. Let $p:W\to Y,q:W\to Z$ denote the two projections. Then by (derived) base change we can identify $i^!j_*O_Z$ with $p_*q^! O_Z$. So finally we summarize:

$$Hom(i_*O_Y, j_*O_Z)= Hom(O_Y, i^!j_*O_Z)= \Gamma_Y(i^!j_*O_Z)=\Gamma_Y(p_*q^! O_Z)=\Gamma_W(q^!O_Z).$$

So we find global sections (again I mean the derived version, i.e., cohomology) of the restriction with supports of $O_Z$ to the (derived) intersection -- i.e., local cohomology of the intersection with coefficients in functions on $Z$. Maybe there's a nice way to say this more intuitively. Note for yet another variant that if you consider $Ext(O_Y,\omega_Z)$ (replace structure sheaf of $Z$ by its dualizing sheaf) then the same calculation yields $\Gamma_W(\omega_W)$, global "top-forms" on the derived intersection. (I guess the Tor version is again an Ext with $\omega_Y$ instead of $O_Y$.)

1

If you replace Ext by Tor, you are defining functions on the derived intersection of $Y$ and $Z$, let's call it $W=Y\times_X Z$ (this is the fiber product in the world of derived schemes, which by definition corresponds to this Tor).

One can then describe the (first) Ext group you asked about as almost functions on the derived intersection of Y and Z (for closed subvarieties). Let's write $i,j$ for the inclusions of $Y$ and $Z$ (yes I know $j$ is usually reserved for open immersions, but anyway..). Then the derived Hom (Ext complex) $Hom(i_*O_Y, j_* O_Z)$ can be calculated by Grothendieck duality for the proper map $i$ as $Hom(O_Y, i^!j_* O_Z)$. Let $p:W\to Y,q:W\to Z$ denote the two projections. Then by (derived) base change we can identify $i^!j_*O_Z$ with $p_*q^! O_Z$. So finally we summarize:

$$Hom(i_*O_Y, j_*O_Z)= Hom(O_Y, i^!j_*O_Z)= \Gamma_Y(i^!j_*O_Z)=\Gamma_Y(p_*q^! O_Z)=\Gamma_W(q^!O_Z).$$

So we find global sections (again I mean the derived version, i.e., cohomology) of the restriction with supports of $O_Z$ to the (derived) intersection -- i.e., local cohomology of the intersection with coefficients in functions on $Z$. Maybe there's a nice way to say this more intuitively. Note for yet another variant that if you consider $Ext(O_Y,\omega_Z)$ (replace structure sheaf of $Z$ by its dualizing sheaf) then the same calculation yields $\Gamma_W(\omega_W)$, global "top-forms" on the derived intersection. (I guess the Tor version is again an Ext with $\omega_Y$ instead of $O_Y$.)