data rate in digital communications
Spreading refers to spreading of the frequency spectrum of the signal, and is achieved by dividing up the symbol interval $T_s$ into many shorter intervals called chip intervals ($L_c$ chip intervals in your notation) and 


transmitting successive bits of the spreading sequence ($L_c$ bits long, remember?) in these $L_c$ chip intervals if the data bit you want to transmit in the symbol interval under consideration happens to be a $0$
transmitting the complements of the successive bits of the spreading sequence in these $L_c$ chip intervals if the data bit you want to transmit in the symbol interval under consideration happens to be a $1$


Symbolically, if $L_c = 3$ and the spreading sequence is $011$, then in a $T_s$-second symbol interval, you get to transmit either $011$ or $100$ in that $T_s$-second interval which we have mentally divided into $3$ subintervals of length $T_s/3$ each. So, the bits transmitted during each of these chip intervals are of shorter duration (by a factor of $3$) and thus require more bandwidth (by a factor of 3 too!) to transmit than we would need if we were transmitting just one data bit every $T_s$ seconds.

"But, but, but,..." you splutter, "I don't have data bits, but rather these beautifully shaped RRC pulses that are of duration $T_s$ seconds. How do I create a CDMA signal from these?"  Well, you have to go back to the purely digital domain where bits are bits, use a frequency multiplier on your clock to get clock pulses at a rate of $L_c/T_s$ Hz so that you can replace each data bit at the $1/T_s$ Hz clock rate with $L_c$ chip bits at the $L_c/T_s$ Hz   clock rate, and tell the analog guys to design their circuits to produce RRC pulses of duration $T_s/L_c$ to accommodate your higher rate digital input signal that is going to send them bits at rate $L_c/T_s$ Hz.