What does the flag formula symbols mean in AVR assembler documentations
To start with the formulas suffer from a typographical error on that HTML version of the documents (the pdf doesn't). The formula you quote should read:

$$\mathrm{Rd3}\cdot\mathrm{Rr3} + \mathrm{Rr3}\cdot\overline{\mathrm{R3}} + \overline{\mathrm{R3}} \cdot\mathrm{Rd3}$$

In that formula there are three values. \$\mathrm{Rd}\$ is the current value of the destination register, \$\mathrm{Rr}\$ is the current value of the source register, and \$\mathrm{R}\$ is the result of the instruction (e.g. for ADD it would be  \$\mathrm{R}=\mathrm{Rd}+\mathrm{Rr}\$).

The number next the value represents which bit from the value that is used in the calculation. In this case it would be bit \$3\$ for each of the registers.

The operators are the standard Boolean operators for AND (\$\cdot\$) and OR (\$+\$).

If you add the inferred brackets (AND has precedence over OR in Boolean equations), you get:

$$(\mathrm{Rd3}\cdot\mathrm{Rr3}) + (\mathrm{Rr3}\cdot\overline{\mathrm{R3}}) + (\overline{\mathrm{R3}} \cdot\mathrm{Rd3})$$

Hopefully in that form it is quite easy to follow.