At the beginning of the proof(theorem 11.10 page 128), it says that it follows easily that the diagonal class can be expressed as r-fold sum \begin{align*} u''=b_i\times c_i+\ldots+b_r\times c_r \end{align*} by using the Kunneth formula \begin{align*} H^*(X\times Y)\cong H^*(X)\otimes H^*(Y) \end{align*} where $b_i,c_i$ are basis for $ H^*(M) $ and $|b_i|+|c_i|=n=\dim(M)$. $u'':=u'|_{M\times M}$ with
$u'\in H^{n}(M\times M,M\times M-\Delta(M))\cong H^{n}(TM,TM-M)$
Here the isomorphism is derived by the diagonal embedding $\Delta:M \rightarrow M\times M\quad x\mapsto (x,x)$ and Tubular Neighborhood Theorem
I think I miss something here, I only know that $H^n(M \times M)=\oplus_{i+j=n}H^{i}(M)\otimes H^{j}(M)$. So i don't know why $u''$ has this explicit form instead of $u''=b_1\times c_1$ or some other formula.
The answer below pointed out my flow. Thanks a lot.
They don't claim that $\{c_1, \dots, c_r\}$ is a basis for $H^*(M)$, they only assume $\{b_1, \dots, b_r\}$ is a basis for $H^*(M)$.
Any element of $H^*(X)\otimes H^*(Y)$ can be written as $\sum_{i=1}^pu_i\otimes v_i$ where $u_1, \dots, u_p \in H^*(X)$ and $v_1,\dots, v_p \in H^*(Y)$. The isomorphism $H^*(X)\otimes H^*(Y) \to H^*(X\times Y)$ is generated by $x\otimes y \mapsto \pi_1^*x\cup\pi_2^*y = x\times y$, so $\sum_{i=1}^pu_i\otimes v_i \mapsto \sum_{i=1}^pu_i\times v_i$. That is, every element of $H^*(X\times Y)$ can be written as $\sum_{i=1}^pu_i\times v_i$ for some $u_1, \dots, u_p \in H^*(X)$ and $v_1,\dots, v_p \in H^*(Y)$.
Now consider the case $X = Y = M$ and suppose $\{b_1, \dots, b_r\}$ is a basis for $H^*(M)$. Then for any $u_i \in H^*(M)$, we have $u_i = \sum_{j=1}^rm_{ij}b_j$ where $m_{ij} \in \Lambda$, the field of coefficients. Therefore
\begin{align*} \sum_{i=1}^pu_i\times v_i &= \sum_{i=1}^p\left(\sum_{j=1}^rm_{ij}b_j\right)\times v_i\\ &= \sum_{i=1}^p\sum_{j=1}^rm_{ij}b_j\times v_i\\ &= \sum_{j=1}^r\sum_{i=1}^pm_{ij}b_j\times v_i\\ &= \sum_{j=1}^r\sum_{i=1}^pb_j\times m_{ij}v_i\\ &= \sum_{j=1}^rb_j\times\left(\sum_{i=1}^pm_{ij}v_i\right)\\ &= \sum_{j=1}^rb_j\times c_j \end{align*}
where $c_j := \sum_{i=1}^pm_{ij}v_i$.