Why superstring theory needs $9+1$ spacetime dimensions is indeed a very good and fundamental question to ask. Unfortunately, it is also very hard to answer this question using only intuitive layman arguments.

The culprit is the concept of a (quantum mechanical) anomaly. The presence of anomalies would render the quantum version of a classical$^{1}$ theory mathematically inconsistent. It turns out that anomaly cancellation conditions for (quantum) string theory are extremely restrictive, and one of their consequences are that flat-spacetime-solutions of (perturbative, quantum) superstring theory must be $9+1$ dimensional.

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$^{1}$ The term classical theory here means a theory where Planck's constant $\hbar=0$ is zero. A classical version of string theory can live in any spacetime dimension.

Why superstring theory needs $9+1$ spacetime dimensions? is indeed a very good and fundamental question to ask. Unfortunately, it is very hard to answer this question using only intuitive layman arguments.

The culprit is the concept of a (quantum mechanical) anomaly. In general, the presence of anomalies would render the quantum version of any classical theory$^{1}$ mathematically inconsistent. 

It turns out that the anomaly cancellation conditions for (quantum) string theory are extremely restrictive. One of their consequences are that flat-spacetime-solutions of (perturbative, quantum) superstring theory must be $9+1$ dimensional.

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$^{1}$ The term classical theory here means a theory where Planck's constant $\hbar=0$ is zero. The classical version of string theory can live in any spacetime dimension.