Why is each observation in a sample considered a random variable in linear regression? I have the following excerpt in my statistics textbook:

I am confused by the sentence: "Another way statisticians treat this model is that, assume $X_1...X_n$ are random variables, we make inferences conditional on their observed values."
Aren't $X_1...X_n$ simply observations of a random variable, $X$? For example let's say I have a random variable $X$ is a persons weight and random variable $Y$ is a persons blood pressure. Then $(X_1, Y_1)$ are an observation of those RVs. How could $X_1$ be a RV?
Maybe its an RV if that specific persons weight (person associated with $X_1$) is sampled multiple times?
 A: 
Aren't $X_1...X_n$ simply observations of a random variable, $X$?

NO.  But this is a good question, as this certainly mystifies many ...  and intuitive language such as observations of a random variable isn't really that helpful.
First, you need to understand what is a random variable, see some of the many posts discussing this:

*

*What is the difference between a realisation of a random variable and random variable itself?


*What is meant by a "random variable"?
(site search will give more). A random variable is a function. And, a mathematical function have only one value.  So it does not make mathematical sense to speak of $X_1, X_2, \dotsc, X_n$ being different values of (or realizations of) one random variable $X$.
Each $X_i$ must be defined as a random variable itself. In terms of your example, where $X$ is a person's weight, and there is a sample of $n$ persons, then $X_1$ is weight of first person in sample, $X_2$ is weight of second person in sample, and so on.
Maybe not that central to your question, but you say confused by also

we make inferences conditional on their observed values

For that, see the following posts,

*

*What is the difference between conditioning on regressors vs. treating them as fixed?


*What are the differences between stochastic and fixed regressors in linear regression model?
