Meaning of Real and Imaginary part of Fourier Transform of a signal
The real and imaginary parts of the Fourier transform of a signal $x(t)$ are the Fourier transforms of the signal's even and odd parts, respectively:

$$X_R(\omega)=\frac12[X(\omega)+X^*(\omega)]\Longleftrightarrow\frac12[x(t)+x^*(-t)]=x_e(t)\\
X_I(\omega)=\frac{1}{2j}[X(\omega)-X^*(\omega)]\Longleftrightarrow\frac{1}{2j}[x(t)-x^*(-t)]=-j\cdot x_o(t)
$$

where $X_R(\omega)$ and $X_I(\omega)$ are the real and imaginary parts of $X(\omega)$, and $x_e(t)$ and $x_o(t)$ are the even and odd parts of $x(t)$, respectively.