I pursue only intuition; please do not answer with formal proofs. I already know the theoretical reason: because each percentage expresses a different base. $1.$ But why not intuitively?
My problem: Whenever adding percentages, I am always initially tempted to add them as cardinal numbers, before resisting myself and spending $\geq 5$ minutes recollecting the following algebra and surmounting the temptation, all of which reveal chasms in my comprehension.
$\bbox[5px,border:2px solid gray]{ \text{ Optional Reading and Supplement: } }$
If the price of apricots ($a$) increases by $p_a$ and the price of cherries ($c$) increases by $p_c$, where $0 \le p_a,p_c \leq 1$; then the $\color{green}{ \text {Correct new price =} (1 + p_a)a + (1 + p_c)c. \tag{2}}$ But adding the percentages as cardinal numbers produces the: $\color{darkred}{ \text { Incorrect new price =} (1 + p_a + p_c)( a + c) = \color{green}{(1 + p_a)a + (1 + p_c)c} \color{#FF4F00}{ + p_ac + p_ca.} \tag{3}}$
The existence of the 2 orange terms proves $2 \neq 3$, but do not reveal the intuition.
PS: This question is motivated by the first sentence of this quote in this question.
This answer is based on the premise that the motivation behind the question is to explain the fallacy described in this question on philosophy.SE.
One point of confusion is the use of the word price. In general, when we must spend money on several resources (flour, labor, utilities, and rent) to make a loaf of bread, the price of a loaf is not determined by the prices of the resources used in its production. Rather, it is determined by the cost of those resources, which is determined by both the price of each resource and the quantity of each resource consumed in making the loaf.
For example, if rent is $\$2000$ per month, that does not mean the baker must charge more than $\$2000$ for one loaf in order to make a profit. Rather, if the bakery produces $10000$ loaves each month, the baker could accurately say that each loaf cost $\$0.20$ of rent to produce, in addition to other costs.
So, assuming the only things the baker has to pay for are for flour, labor, utilities, and rent, the baker's cost to make a loaf of bread is
$$ B = F \cdot Q_F + L \cdot Q_L + U \cdot Q_U + R \cdot Q_R, $$
where $F$, $L$, $U$, and $R$ are the prices of flour, labor, utilities, and rent, and $Q_F$, $Q_L$, $Q_U$, and $Q_R$ are the quantities of each of those resources that the baker has to expend on each loaf.
Now, assuming the amount of each resource per loaf remains constant, it is true that a $10$ percent increase in the price of one resource corresponds to a $10$ percent increase in the cost of that resource. For example, if the price of flour goes up $10$ percent, from $F$ to $(1 + 0.10)F$, the cost of flour per loaf also increases $10$ percent, because $$ ((1 + 0.10)F) \cdot Q_F = (1 + 0.10)(F \cdot Q_F). $$
Of course, if we just say "cost" instead of price when describing the baker's expenses, we bypass all of these complications. So let's do that.
What happens, then, when the costs of all these items increase? The baker was writing checks to four different groups of people before -- the flour merchants, his laborers, the utility companies, and his landlord -- and he is still writing checks to the same four groups of people. The amount of each group of checks has merely increased. So if the baker was paying $\$600$ to the flour merchants each month before, and flour goes up $10$ percent, now he must pay the flour merchants $\$660$ each month. If the baker was previously paying his laborers a total of $\$5000$ per month, and labor rates increase $20$ percent, he must now pay his laborers $\$6000$ per month. That is, for flour and labor, the new costs add up to
$$ (1 + 0.10) 600 + (1 + 0.20) 5000, $$
which are analogous to the terms $(1 + p_a)a + (1 + p_c)c$ if we say that $a$ and $c$ were the previous total costs of apricots and cherries we bought, rather than their unit prices. Similarly, if utilities and rent were $\$200$ and $\$2000$, respectively, and these go up $10$ percent each, the baker's total costs after all these increases are
\begin{align} (1 + 0.10) 600 + (1 + 0.20) 5000 + (1 + 0.10) 200 + (1 &+ 0.10) 2000 \\ &= 660 + 6000 + 220 + 2200 \\ &= 9080. \end{align}
There are no other added costs. When we say labor increased $20$ percent, that $20$ percent accounts for all of the extra money the baker must pay his laborers, and likewise the percentage increases in each other cost account for all the extra money the baker must pay each of the other three groups of people.
Given all of that, what would it mean to take the $10$-percent increase in flour and multiply it by the cost of labor? This would be $0.10 \cdot \$5000 = \$500$ in our example; but to whom does the baker pay this extra $\$500$?
If each of the baker's costs individually had gone up $50$ percent, then indeed his cost per loaf would also have increased $50$ percent. Here's how. Before the increase, the baker's total costs were
$$ 600 + 5000 + 200 + 2000 = 7800.$$
That's $\$0.78$ per loaf (since the bakery produces $10000$ loaves). After $50$-percent increases in flour, labor, utilities, and rent, all occurring in the same month, the baker's costs would be
$$ 900 + 7500 + 300 + 3000 = 11700,$$
which is $1.17$ per loaf, which is indeed a $50$-percent increase. But of course that's not what happened at all, and the baker either is not paying attention to his own accounts (or is trying to fool us) when he says it is what happened.