Risk.
Fama-French's factor selection for the 3 and 5 factor models is premised on the idea that systemic returns above or below a benchmark are due only to the assumption of risk (i.e., investors who overweight risky (e.g., under-regarded and out of favor) securities only do so in anticipation of greater rewards).
The history of this core intuition dates back to Modigliani Miller (I or II) in which the authors proposed a) leverage does not change the value of a firm (or its expected returns on capital), and b) excess returns could be explained by a beta coefficient which represented risk. The CAPM then proposed that this beta could be directly observed from market returns and covariances. FF then retorted with the FF 3 factor model (1992) to demonstrate that such a simple explanation for risk misrepresented the core beliefs about market efficiency.
Under the Chicago School, market efficiency is not a belief that market prices are right, but rather than it is exceedingly difficult to determine if they are wrong. In an Chicagoan efficient market, in which investors allocate capital according to their beliefs and tolerance for risk, beliefs are still fallible. Therefore, I don't actually think FF intended their factor models for use in tactical asset allocation or to explain all variation in returns, but rather to demonstrate the shortcomings of CAPM.
As such, it is important to note there is no such thing as "cheap" in the FF universe. For example, the B/M factor (HML) is not meant to convey "cheapness", but rather investors' expectations regarding a firm's future ability to pay investors. This intuition, in turn, comes from the dividend discount model (DDM) in which the net present value is defined by the discounted future cash flows which investors receive from holding a particular security or basket of securities.
FF (2015) elaborate:
There is much evidence that average stock returns are related to the
book-to-market equity ratio, B/M. There is also evidence that
profitability and investment add to the description of average returns
provided by B/M. The logic for why these variables are related to
average returns can be explained via the dividend discount model. The
model says that the market value of a share of stock is the present
value of expected dividends per share,
(1) $$ m_t = \sum_{\tau=1}^{\infty} E \left(d_{t+1} \right)\frac{1}{1+r^\tau} $$
In this equation $m_t$ is the share price at time $t$, $E(dt+τ)$ is the
expected dividend per share in period $t+τ$, and $r$ is (approximately)
the long-term average expected stock return or, more precisely, the
internal rate of return on expected dividends.
Therefore, firms with relative low book values relative to market values imply a) high future earnings and/or, b) low risk. Likewise, firm with relatively high book values relative to market values imply a) low future earnings, and/or b) high risk.
The profitability factor (i.e., operating return on equity) is related to the DDM. Incorporating a proxy measure for net present value which is distinct from book value helps to differentiate between marginal and robust firms. On first appearance, the profitability does not fit FF's theme on risk. However, it can be seen as a proxy for book value's sensitivity to risk.
Likewise, the intuitions for the selecting size (SMB) and investment (i.e., asset growth) factors relate back to risk. With size, it is generally regarded that smaller business are riskier than large ones. With investment, firms which invest are generally as safe, while firms which underinvest are generally regarded as risky.
Notably, momentum does not fit into the broader framework. Although, FF continue to track portfolios constructed by sorts of prior returns, momentum is not formally considered as evidence for or against the EMH. Rather, it is an anomaly which might be better explained by the behavioral economists. Mean reversion, on the other hand, is not as problematic since it is presumably caused by timing issues in supply/demand and is expensive to trade.
