Calculate the total fraction of salary spent on food and rent.

$$\text{Fraction spent on food and rent} = \frac{1}{3} + \frac{2}{7}$$

To add these fractions, find a common denominator, which is $3 \times 7 = 21$.

$$\frac{1}{3} + \frac{2}{7} = \frac{1 \times 7}{3 \times 7} + \frac{2 \times 3}{7 \times 3} = \frac{7}{21} + \frac{6}{21} = \frac{7+6}{21} = \frac{13}{21}$$

$$\text{Remainder} = 1 - \text{Fraction spent on food and rent}$$ $$\text{Remainder} = 1 - \frac{13}{21} = \frac{21}{21} - \frac{13}{21} = \frac{21-13}{21} = \frac{8}{21}$$

The employee uses $\frac{1}{4}$ of the remainder on transport.

$$\text{Fraction spent on transport} = \frac{1}{4} \times \text{Remainder}$$ $$\text{Fraction spent on transport} = \frac{1}{4} \times \frac{8}{21} = \frac{1 \times 8}{4 \times 21} = \frac{8}{84} = \frac{2}{21}$$

The employee saves the rest of the remainder after transport. If $\frac{1}{4}$ of the remainder is spent on transport, then $1 - \frac{1}{4} = \frac{3}{4}$ of the remainder is saved.

$$\text{Fraction saved} = \frac{3}{4} \times \text{Remainder}$$ $$\text{Fraction saved} = \frac{3}{4} \times \frac{8}{21} = \frac{3 \times 8}{4 \times 21} = \frac{24}{84}$$

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 12.

$$\frac{24 \div 12}{84 \div 12} = \frac{2}{7}$$

The fraction of his salary he saves is $\frac{2}{7}$.