I'm not quite sure what kind of answer you're expecting, but here is a geometric example that may help to grasp some intuition.

In differential geometry, when an intersection is badly behaved (e.g. it doesn't have the expected dimension) one can geometrically perturbe one of the two factors. For instance, if you are intersecting tow submanifolds $X,Y\subset Z$, and if $X$ is locally given as the zero of some functions $X\overset{loc}{=}\{f_1=\dots=f_k=0\}$, you may want to introduce a deformation $X_{t_1,\dots,t_k}$ of $X$ defined as $$ X_{t_1,\dots,t_k}\overset{loc}{=}\{f_1=t_1,\dots,f_k=t_k\} $$ One of the main idea of derived geometry is to replace these deformation/perturbation parameters $t_i$'s by a homological perturbation: $$ X_{t_1,\dots,t_k}\overset{loc}{=}\{f_1=d\tau_1,\dots,f_k=d\tau_k\}, $$ with $deg(\tau_i)=-1$ (my degree convention is cohomological).

Homological perturbation has two advantages above geometric perturbations:

• it can be made functorial.

• it exists even in the (quite non-flexible) algebraic setting, where geometric perturbation may not exist.

Let's try to applyi informally the above reasonning to the case discussed in Jon Pridham's comment: consider $X=Y=\{x=0\}$ inside $Z=\mathbb{A}^1=Spec(k[x])$. You deform $\{x=0\}$ to $\{x=d\tau\}$ and then proceed with the intersection of $\{x=d\tau\}$ with $\{x=0\}$, and get $\{d\tau=0\}$, which is just a ($k$-)point (it is $0$ in $\mathbb{A}^1$) together with a self-homotopy (given by $\tau$). This is indeed the "space" of derived loops in the affine line that are based at $0$.

I apologize for self-promoting, but you can read an informal account of how to view derived self-intersections as some kind of based loop spaces in the introduction of https://arxiv.org/pdf/1306.5260.pdf.

I'm not quite sure what kind of answer you're expecting, but here is a geometric example that may help to grasp some intuition.

In differential geometry, when an intersection is badly behaved (e.g. it doesn't have the expected dimension) one can geometrically perturbe one of the two factors. For instance, if you are intersecting tow submanifolds $X,Y\subset Z$, and if $X$ is locally given as the zero of some functions $X\overset{\text{loc}}{=}\{f_1=\dotsb=f_k=0\}$, you may want to introduce a deformation $X_{t_1,\dotsc,t_k}$ of $X$ defined as $$ X_{t_1,\dotsc,t_k}\overset{\text{loc}}{=}\{f_1=t_1,\dotsc,f_k=t_k\}. $$ One of the main idea of derived geometry is to replace these deformation/perturbation parameters $t_i$'s by a homological perturbation: $$ X_{t_1,\dotsc,t_k}\overset{\text{loc}}{=}\{f_1=d\tau_1,\dotsc,f_k=d\tau_k\}, $$ with $\operatorname{deg}(\tau_i)=-1$ (my degree convention is cohomological).

Homological perturbation has two advantages above geometric perturbations:

• it can be made functorial.

• it exists even in the (quite non-flexible) algebraic setting, where geometric perturbation may not exist.

Let's try to apply informally the above reasoning to the case discussed in Jon Pridham's comment: consider $X=Y=\{x=0\}$ inside $Z=\mathbb{A}^1=\operatorname{Spec}(k[x])$. You deform $\{x=0\}$ to $\{x=d\tau\}$ and then proceed with the intersection of $\{x=d\tau\}$ with $\{x=0\}$, and get $\{d\tau=0\}$, which is just a ($k$-)point (it is $0$ in $\mathbb{A}^1$) together with a self-homotopy (given by $\tau$). This is indeed the "space" of derived loops in the affine line that are based at $0$.

I apologize for self-promoting, but you can read an informal account of how to view derived self-intersections as some kind of based loop spaces in the introduction of Calaque, Căldăraru, and Tu - On the Lie algebroid of a derived self-intersection.