What exactly happens with Spread Spectrum Modulation
Direct-sequence spread spectrum (DSSS) is a technique that is used to generate a modulated signal that occupies more bandwidth than would be implied by its information content alone. A DSSS signal is generated via multiplication of a (typically digitally-modulated) baseband signal by another spreading code waveform. The spreading code waveform is constructed such that it is modulated at a higher rate than the baseband signal; this yields the "spreading" of the resulting signal's spectrum. 

To see this, recall the multiplication property of the Fourier transform. Let the baseband signal $x(t)$ be multiplied by a spreading waveform $s(t)$:

$$
x(t)s(t)\Longleftrightarrow X(f) * S(f)
$$

where the $*$ denotes convolution. If $S(f)$ is broadband when compared to $X(f)$, then the result of convolving $X(f)$ with $S(f)$ is that $X(f)$'s spectral content is "smeared" or "spread" across a wider region of the spectrum. The amount of spreading is dependent upon $S(f)$ and therefore the choice of the spreading sequence $s(t)$. So, for example, you could take a 1 kbps BPSK signal, which would normally have a null-to-null bandwidth of 2 kHz, and spread it to instead have a null-to-null bandwidth of your choice: say, 1 MHz. 

To answer your most pointed question, you're not really sending a single signal redundantly on multiple carrier frequencies, nor are you dividing the input signal into pieces that get transmitted at multiple carriers. In lieu of a discrete multi-carrier approach, DSSS instead is more of a continuous smearing of the input signal over a large region of spectrum.

If the receiver is also cognizant of the spreading sequence $s(t)$, then it is capable of despreading its observation of the transmission and therefore recovering the original baseband signal. As long as the spreading sequence $s(t)$ does equal zero for any $t$, then in principle, the spreading process is perfectly reversible. In practice, this isn't exactly the case, as there will be some distortion between what is transmitted and what the receiver sees, but it leads to an important point that is often misunderstood about DSSS:


  No benefit in bit error performance, measured in $\frac{E_b}{N_0}$ (the typical metric used for digital modulation performance evaluation) is offered by using direct-sequence spread spectrum. Since the spreading process is reversible, then the principle of reversibility states that there can be no improvement in performance of the optimal receiver for the spread signal when compared to the performance of the optimal receiver for the original, unspread signal.


So if there's no direct bit-error performance benefit, why would you want to do this? There are a few common reasons:


Multiple access: Code division multiple access schemes work using DSSS techniques. CDMA allows multiple transmitted signals to occupy the same bandwidth over the same time period. Transmitters are associated with separate carefully-selected spreading codes; these are also known by the receiver, which uses them to separate the previously-intermingled-in-time-and-frequency signals. CDMA can be made to seem like the greatest thing since sliced bread, but there are some practical issues (most notably the near-far problem) that limit its usefulness. It is, however, used in some cellular networks around the world, for one example.
Interference mitigation: Signals with high bandwidth are resistant to narrowband interferers on the AWGN channel. A hand-waving time-domain explanation: a wideband signal in the frequency domain will have a narrow autocorrelation function in the time domain, and vice versa. The optimal receiver will use a correlator (often implemented with a matched filter) matched to the transmitted spread-spectrum waveform, so its output (which ideally will contain copies of the signal's autocorrelation function at symbol-spaced intervals) will contain narrow, sharp peaks that will be easy to identify against the oppositely-characterized background interference.

A hand-waving frequency-domain explanation: the spreading process takes the original baseband signal's power and spreads it across a wide region of the spectrum. In the communication channel, narrowband interference finds its way into the signal that eventually is observed. During the despreading process, the DSSS signal is compressed back to the original baseband signal's bandwidth. However, at the same time, the despreading process actually spreads the narrowband interference across the same bandwidth as the original DSSS signal. 

So effectively, the narrowband interferer has been transformed into a wideband interferer across the entire DSSS signal bandwidth, which as you recall, is much larger than the baseband (information-carrying) signal's bandwidth. Therefore, much of the interference power is pushed out of band with respect to the signal of interest, which yields a higher signal-to-interference power ratio (and better performance).
"Discreetness": No, not discrete, discreet. DSSS is often spoken of as being able to "pull signal out of the noise floor." This refers to the ability of a spread-spectrum receiver to transform a wideband DSSS signal back to the original, much narrower waveform. For a given transmitter power, as the transmitted spread signal bandwidth increases, the average power spectral density over the DSSS bandwidth will decrease, perhaps to the point where it is difficult to observe using spectral analysis at the receiver. This property can be useful for a number of reasons. 

One aim might be to decrease the probability of the emitter being detected (and possibly intercepted) by an adversary. Another might be to mitigate interference with other, narrowerband systems that operate in the same region of spectrum. An example of this approach is in ultra wideband communications systems, which have increased in popularity in recent years.